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Molecular Mechanics • Studies involving covalent interactions (enzyme reaction): quantum mechanics; extremely slow • Studies involving noncovalent interactions (conformational references, molecular recognition): classical mechanics; acceptable for a few structures • Studies involving sequences only: statistical formalisms; extremely fast Molecular Mechanics • Study how protein/protein, protein/ligand, protein/NA interactions. Why they are specific? how to mimic them? • Use them in structure-based drug design, docking. • Study how proteins/NAs change conformations. How a specific function/mechanism is realized? Theoretical Ground: Classical Mechanics Building on the work of Galileo and others, Newton unveiled his laws of motion in 1686. According to Newton: • I. A body remains at rest or in uniform motion (constant velocity - both speed and direction) unless acted on by a net external force. • II. In response to a net external force, F, a body of mass m accelerates with acceleration a = F/m. • III. If body i pushes on body j with a force Fij, then body j pushes on body i with a force Fji. Theoretical Ground: Classical Mechanics • How to obtain forces? Easy if an energy model is given. Where to use Molecular Mechanics Energy Model? • Molecules containing thousands of atoms • Organics, oligonucleotides, and peptides • Vacuum, implicit, or explicit solvent environments • Ground state only • Thermodynamic and kinetic via simulations. Building Principles of Molecular Mechanics (Energy Model) • Nuclei and electrons are lumped into atom-like particles • Atom-like particles are spherical (radii obtained from measurements or theory) and have a net charge (obtained from theory) • Interactions are based on springs and classical potentials • Interactions must be preassigned to specific sets of atoms • Interactions determine the spatial distribution of atom-like particles and their energies Simplistic Molecular Mechanics Force Field Bond Improper Dihedral Angle Dihedral Van der Waals Charge - Charge Bond Stretching Energy Bond Stretching Energy Angle Bending Energy Angle Bending Energy Significance of Energy Parameters Torsion Energy The torsion energy is modeled by a simple periodic function: Significance of Energy Parameters The Roles of Torsion Energy • The torsion energy in molecular mechanics is primarily used to correct the remaining energy terms rather than to represent a physical process. • The torsional energy represents the amount of energy that must be added to or subtracted from the Stretching + Bending + Non-Bonded interaction terms to make the total energy agree with experiment or rigorous quantum mechanical calculation for a model dihedral angle (ethane, for example might be used a model for any H-CC-H bond). Cross Terms Possible cross terms: • stretch-stretch, stretch-bend, strechtorsion; • bend-bend, bend-torsion; • torsion-torsion. (Fig. 4.13, Leach) Needed in studies of high-frequency motions, i.e. vibrational spectra. Non-Bonded Energy Van der Waals Energy Significance of Energy Parameters Electrostatic Energy • The electrostatic contribution is modeled using a Coulombic potential. • The electrostatic energy is a function of: o (a) charges on the non-bonded atoms; o (b) inter-atomic distance; o (c) molecular dielectric expression that accounts for the attenuation of electrostatic interaction by the molecule itself. Electrostatic Energy: Dielectrics • The molecular dielectric is set to a constant value between 1.0 and 4.0. However, it has to be consistent with how a force field is designed. (not a free parameter) • A linearly varying distance-dependent dielectric (i.e. 1/r) is sometimes used to account for the increase in the solvent (aka, water) dielectrics as the separation distance between interacting atoms increases. (This is being abandoned) • When it is needed, the Poisson’s equation, or its approximation, has to be used. (This is gaining popularity) Other Nonbonded Interactions: Hydrogen Bonding • Hydrogen bonding term is usually wrapped into the electrostatic term in force fields widely used today. However it does not imply that hydrogen bonding is purely electrostatic in nature. • Hydrogen bonding, if explicitly represented, uses a 10-12 Lennard-Jones potentials. This replaces the 6-12 Lennard-Jones term for atoms involved in hydrogen-bonding. Other Nonbonded Interactions: Polarization • Polarization is important when large environmental changes occur, i.e. from protein interior to water, or from membrane to water. • Usually modeled as inducible dipole: μ = aE • Note it is not free to induce a dipole: the work done is 1/2 aE2. • Finally, electrostatic energy includes charge-charge, charge-dipole, and dipole-dipole; or electrostatic field is from charge and dipole. • No stable force fields with polarization available right now! Scaling of Nonbonded Terms • Scaling of electrostatic energy: chargecharge 1/r; charge-dipole 1/r2, dipoledipole 1/r3. • Scaling of van der Waals energy: 1/r6. • The example of two point charges on zaxis.